On Spatially Homogeneous Solutions of a Modified Boltzmann Equation for Fermi–Dirac Particles

The paper considers a modified spatially homogeneous Boltzmann equation for Fermi–Dirac particles (BFD). We prove that for the BFD equation there are only two classes of equilibria: the first ones are Fermi–Dirac distributions, the second ones are characteristic functions of the Euclidean balls, and they can be simply classified in terms of temperatures: T>2/5TF and T=2/5TF, where TF denotes the Fermi temperature. In general we show that the L∞-bound 0≤f≤ 1/ε derived from the equation for solutions implies the temperature inequality T≥2/5TF, and if T>2/5TF, then f trend towards Fermi–Dirac distributions; if T=2/5TF, then f are the second equilibria. In order to study the long-time behavior, we also prove the conservation of energy and the entropy identity, and establish the moment production estimates for hard potentials.